Question

use reasoning to solve a problem
example 4 put the numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9 in the circles so that each edge adds up to the same number.

Explanation:

Step1: Calculate sum of 1 - 9

The sum of numbers from 1 to 9 is given by the formula $\frac{n(n + 1)}{2}$, where $n=9$. So, $\frac{9\times(9 + 1)}{2}=45$.

Step2: Analyze number of edges

Assume the figure has 3 edges (a common - type of such number - placement puzzles like a triangle - based one). Let the sum of each edge be $x$. The sum of all edges will count some numbers more than once.

Step3: Try combinations

By trial - and - error, if we consider a triangular arrangement of circles:
Let the vertices of the triangle be $a$, $b$, $c$. Each edge sum is $x=a + d+e=b + f+g=c + h + i$. The sum of all three edges is $3x$. The sum of all 9 numbers is 45. And the vertices are counted twice.
After trying different combinations, one possible arrangement for a triangular - like figure (where each edge has 3 non - overlapping circles except at vertices):
One edge: 1, 9, 5; another edge: 2, 7, 6; third edge: 3, 4, 8. Each edge sums to 15.

Answer:

One possible solution (for a triangular - like arrangement): One edge has 1, 9, 5; another has 2, 7, 6; the third has 3, 4, 8 (there are other valid arrangements as well).